Dear student,

Newton backward interpolation is very helpful tool for finding the in between data and with this we can predict the past datas.

In newton forward interpolation we use forward difference operator and in newtons backward interpolation we use backward different operator.
Since making the sign of backward difference operator will be difficult so, i will be assuming it as " bdo "

Now in Newton's backward interpolation we have :-

f(X) = f(Xn + hu)
u = (X- Xn)/h

Therefore the formula for netwon backward interpolation is :-

f(X) = Yn + u(bdoYn) + u(u+1)(bdo^2Yn)/2 + u(u+1)(u+2)(bdo^3Yn)/6 + . . . . . .

2 = 2 factorial
6 = 3 factorials

And so on..

I hope this will help you.